\section{Introduction}
Recently, many people use data packet services by using mobile terminals after 3G (third generation) mobile services such as 'HSDPA' and 'Mobile WiMAX' appeared. Moreover, the evolution of the smart-phone makes consumers want to use data packet services anytime and anywhere. For example, Apple's '3G iPhone', which is a kind of smart-phone, provides music downloads, video streaming, internet and application downloads using the 'HSDPA' technology. However, although consumers desire high-speed data transfer rate even when they are staying in indoor areaa, it is hard to provide a satisfactory service rate because wireless signals from base station becomes weaker through walls. Unfortunately, at least 70 percent of data traffic packets are generated indoors.

In order to solve indoor coverage problem, some research groups such as the 3GPP and the IEEE 802.16 start to consider femto cell topology \cite{CAG08FNS,CHS08OFC,3GPP} which means a very small cell structure in cellular networks. The Fig.\ref{femto} shows an example of the femto cell system. As you can see in the figure, a small femto AP is used to support indoor users. Since the femto APs are deployed in indoor areas and existing nearby users, they could have good channel quality to support high-speed packet transmissions.

%================================================================
\begin{figure}
\centering{\includegraphics[width=2.3in]{fig/femtomodel}}
\caption{Network configuration of femto cell system} \label{femto}
\end{figure}
%================================================================

Although the femto cell solves the coverage problem involving indoor users, there are some remained challenges. One of the main issues is interference problems \cite{CAG08FNS,CHS08OFC,3GPP}. Since many service providers do not have enough frequency resource to provide services, femto cells might use the same channel as that of macro cells. Consequently, when a macro user exists nearby a femto AP, the femto AP's transmission signals interfere with the receiving signal of the macro user. Therefore, in downlink case, due to the signal which is transmitted by femto AP, the data rate of macro user reduces because the SINR becomes lower because of the interference. It is a critical problem from the veiwpoint of service providers since they want to maintain the performance of macro users. Moreover, since the femto AP might allow only notified users to connect so that only the owner of the femto AP could get benefits from the femto AP, non-authorized users nearby the femto AP can not change their serving BS and suffer from significant interference from the femto AP even if the received signal power of the femto AP is larger than that of their current serving BS. Consequently, in some area, macro users can not decode any message from their serving base station. This area is called the "coverage hole". Furthermore, there are not only interference issues between a femto cell and a macro cell but also inter femto cell interference problems. Since a femto AP can be easily deployed by any users, it is hard to control the interference among various femto APs, particularly when they are installed nearby.

In some previous research work made to solve the above-mentioned issues, the power control scheme was exploited to reduce interference \cite{CHS08OFC}, because if a femto AP reduces its transmitted power, the power changing could reduce interference for neighboring users. In \cite{CHS08OFC}, when a new femto AP is turned on, the femto AP estimates received macro cell power and determines its transmission power to limit interference. Therefore, the power control scheme which is proposed by \cite{CHS08OFC} guarantees that a large coverage hole could be avoided.

If we assume that the incoming traffic rate of a femto AP is lower than the channel capacity of the femto AP, we could reduce femto AP's power without any throughput degradation. Fortunately, this assumption is possible for some cases. For example, because there might be one or two users in femto cell when the femto AP is deployed in homes, the femto AP does not need high data rate if the users do not demand lots of packets. Moreover, backhaul link capacity could be less than air medium capacity in real environments \cite{CAG08FNS}. For example, when users of the femto cell watch IPTV or download some large size data to their PC, the femto AP experiences backhaul bottleneck since it shares ethernet link with other services. In that case, the femto AP does not need to operate with full transmission power. Therefore, in this paper, we propose a power control scheme that takes into consideration the traffic density of the femto AP.

The remaining of this paper is organized as follows. Section~\ref{sec:power-contr-accord}
defines the system models. The proposed scheme which is a power control algorithm for a femto AP is explained in section~\ref{sec:system-model}. In section~\ref{sec:performance-analysis}, we explain
an finite state markov chain (FSMC) model to analyze system performance. By using the FSMC model, numerical results are derived and explained in section~\ref{sec:numerical-results}. Finally, we conclude our paper in section~\ref{sec:conclusions-1}.

\hfill

\section{Power control according to traffic density}\label{sec:power-contr-accord}
\subsection{Packet based power control}
In AWGN wireless channel model, the $C_{ij}$ which is the capacity of the link between node i and node j is expressed as 
%===========================================================
\begin{equation}
\label{capacityeq}
C_{ij}=W \log_{2}(1+\frac{|h_{ii}|P_{i}}{N_{0}+\sum_{j\neq i}|h_{ji}|P_{j}})
\end{equation}
%===========================================================
where $h_{ij}$ refers channel gain between node i and node j and $P_{i}$ is transmission power of node i. From Eq.(\ref{capacityeq}), we already know that its own capacity decreases but capcities of other links increase as its transmission power decreases. Thus, power control is very useful for interference reduction though its capacity should be degraded. 

When we consider the power control for a femto AP, there are some issues to solve. One of that is a handoff problem. In general, when received signal power of a mobile node from one of its neighbor base stations is higher than the received signal strength form the serving base staion, the mobile node change its serving base station to the base station transmiting the signal which has the highest received signal strength. So, as the femto AP reduces its transmission power, mobile terminals which are served by the femto AP should try handoff more offently. Additionally, channel variation of neighbor nodes is also an problem which is occured by the power control. Because channel fluctuation could make significant error when they use adaptive channel coding scheme for channel variation, power control scheme of the femto AP might lead throughput degradation of neighbor cells. Therefore, we should consider the problems to sugguest a power control algorithm for femto AP.

%================================================================
\begin{figure}[!t]
\centering{\includegraphics[width=2in]{fig/algorithm}}
\caption{Relationship between queue length and operation mode} \label{algo}
\end{figure}
%================================================================

To construct a power control algorithm that adapatively operates  based on traffic density, we should think about how to estimate the traffic density. In our scheme, the queue length of the femto AP is exploited to estimate the traffic density. Since the queue length implies the femto AP's required service amount, it is reasonable that the power control scheme use the queue length for estimating the traffic density. In order to reduce the channel fluctuation of neighbor nodes by the femto AP's power control, transmission power level should be not changed oftenly. Therefore, the femto AP does not check its queue length every time but checks every $N_L$ frame interval, so that reduce the channel variation of neighbor nodes. When the femto AP checks its queue length, the increase in the queue length means that the serving data rate is less than the input data rate. On the other hand, the decrease in the queue length means that the incoming data rate of the femto AP is less than the serving data rate. Therefore, when the queue length becomes larger, we should increase transmission power to boost packet transmission speed and when the queue length is reduced, we could decrease transmission power to degrade the interference for neighbor users.

To provide a such power control algorithm, the proposed scheme exploit two operation modes, which will be called mode1 and mode2. If a femto AP operates in mode1, the femto AP uses half of its power to transmit and the mode2 means that the femto AP transmits packets with full power. By using the operation mode and some thresholds which are $\alpha$ and $\beta$ in the queue length, the femto AP operates in mode1 if the queue length is less than the $\beta$ and in mode2 if the queue length is larger than the $\alpha$ in our proposed scheme. Becasue the $\alpha$ is larger than $\beta$, the proposed scheme is expected that it reduces interference to neighbors. Moreover, the femto AP broadcasts its operation mode every frame to prevend unwanted handoff. When a node receives the mode message, the node can estimate received signal strength of the femto AP when transmits with full power. Therefore, by using the information, the unwanted handover problem should be solved since the received signal strength of the femto AP could be revised by the message. Fig.\ref{algo} represents the operation of the proposed scheme where $P_f$ is the transmission power of the femto AP.

\subsection{User based power control}
If the variation of traffic density is dramdically, the previous algorithm which is traffic based power control scheme could not make any performance enhancement. 


\hfill


\section{System model}\label{sec:system-model}
%====================================================
\begin{table}[!t]
\renewcommand{\arraystretch}{1.2}
\caption{MCS levels in the IEEE 802.16 system}
\label{mcstable}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
MCS level & packets/frame & AMC mode & SINR thrshold value\\ 
\hline
\hline
1 & 2 & QPSK-1/2 & 6.0 dB\\
\hline
2 & 3 & QPSK-3/4 & 9.0 dB\\
\hline
3 & 4 & 16QAM-1/2 & 12.0 dB\\
\hline
4 & 6 & 16QAM-3/4 & 15.0 dB\\
\hline
5 & 9 & 64QAM-3/4 & 21.0 dB\\
\hline
\end{tabular}
\end{center}
\end{table}
%=====================================================
Our proposed algorithm is based on the IEEE 802.16e standard \cite{IEEE} which is developed to provide high-speed data packet services. However, our proposed scheme might be suitable for any other standards. The IEEE 802.16e system exploits adaptive modulation and coding (AMC) schemes to get throughput enhancement \cite{wimax, dhcho}. Generally, since the AMC schemes change modulation and coding scheme (MCS) level by channel state, a base station which wants to transmit some packets by using the AMC scheme should know channel quality indicator (CQI) such as SINR at its destination node. Therefore, the destination node sends CQI value to the base station and the base station determines its MCS level according to the SINR. In the case of the IEEE 802.16e standard, a mobile terminal calculates its SINR value by a preamble signal \cite{IEEE, dhcho}. After the CQI reporting, the base station makes a decision about MCS level by using SINR-MCS table. Table \ref{mcstable} is an example for the SINR-MCS table in the IEEE 802.16 standard. In this paper, the Table.\ref{mcstable} is used for our modeling and analysis.

We assume that every packet size is the same and packet arrival distribution of a base station follows Poisson distribution. Moreover, we assume that the packet arrival distribution is i.i.d for every frame. Let $A_n$ denote the number of arrival packet at the n-th frame. Then, the distribution of $A_n$ can be represented as
%=============================================================================
\begin{equation}
Pr\{A_{n} = k\}=\frac{\lambda^k e^{-\lambda}}{k!}
\end{equation}
%=============================================================================
where $\lambda$ is the average number of arrival packets at a frame and $k \ge 0$. When the packets arrive, they are stacked on the base station's queue. The queue length is finite and we denote the length of the queue as $Q_{L}$. It means that a base station can't have more than $Q_{L}$ packets at its queue. Therefore, when packets arrive at a base station and the queue of the base station overflows, incoming packets should be dropped.

In this paper, the Nakagami-m model is used for our time-varying channel model. Then, the probability density function (pdf) of the received SINR $\gamma$ is described as
%=============================================================================
\begin{equation}
p_{\gamma}(\gamma) = \frac{m^m \gamma^{m-1}}{\bar{\gamma}\Gamma (m)}e^{- \frac{m \gamma}{\bar{\gamma}}}
\end{equation}
%=============================================================================
where $\bar{\gamma}$ is an average SINR and $\Gamma (x)$ is a gamma function \cite{smith}. However, we don't consider shadowing. For channel modeling, we assume that the channel state is invariant during a frame transmission and that it can be changed frame-by-frame, since indoor channels might be slowly-varying channels. Thus, we can construct a finite state markov chain (FSMC) channel model \cite{LZ05QAM} in this paper. And then, we assume that the femto AP knows perfect CQI at the receiver.


\hfill



\section{Performance analysis}\label{sec:performance-analysis}

In \cite{LZ05QAM}, the authors make a queueing modeling for the MCS scheme. Similar with \cite{LZ05QAM}, we can make an FSMC model for our proposed scheme.


%=================================================================
\begin{table}[!t]
\renewcommand{\arraystretch}{1.2}
\caption{MCS level and data rate for channel state}
\label{chantab}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
State              &1&2&3&4&5&6&7&8&9\\ 
\hline
MCS &               0&0&1&2&3&4&4&5&5  \\
\hline
Packets/frame&      0&0&2&3&4&6&6&9&9 \\
\hline
\end{tabular}
\end{center}
\end{table}
%=================================================================
%================================================================
\begin{figure}[!t]
\centering{\includegraphics[width=1.5in]{fig/channelstate}}
\caption{Channel state transition according to mode change} \label{chanalstate}
\end{figure}
%================================================================
As we mentioned on the system model section, we consider the Nakagami-m fading channel as wireless channel of this paper and we use an FSMC channel model for performance analysis. In order to make the FSMC channel, we have to define channel states. For convenient analysis, we make each channel state have the same MCS level in each SINR range, e.g. if the SINR range of a channel state is from 6dB to 9dB, the channel could have same MCS level 1 from Table. \ref{mcstable}. Moreover, since our proposed scheme changes the femto AP's transmission power by 3dB according to the queue length of the femto AP, we set the SINR distance between adjacent channel states to 3 dB, so that a channel state changes to a particular state when the femto AP changes its transmission power. Fig.  \ref{chanalstate} represents channel states of our modeling. Each of the channel states has the same MCS level as shown in Table. \ref{chantab} and changes to a particular channel state when the femto AP changes the transmission power. The channel state transition probabilities can be obtained by using the equations in \cite{LZ05QAM}.
%================================================================
\begin{figure}[!t]
\centering{\includegraphics[width=3.3in]{fig/queuestate}}
\caption{Relation between queue length, channel state and arrival packets.} \label{asdf}
\end{figure}
%================================================================

Let $Q_{n}$, $C_{n}$ and $A_{n}$ denote the number of packets in the queue, the channel state and the number of arrival packets at time n, respectively, and let $D(C_{n})$ denote the value of packets/frame of the channel state $C_{n}$. The value of packets/frame can be found in Table. \ref{chantab}. As shown in Fig. \ref{asdf}, during n-th frame, the femto AP sends $D(C_{n})$ packets from $Q_{n}$ packets at time n and $A_{n}$ packets arrive in the queue. Consequently, the relationship between $Q_{n}$ and $Q_{n-1}$ can be represented as
%=============================================================================
\begin{equation}
Q_{n} = min\{ Q_{L}, max\{ 0, Q_{n-1} - D(C_{n-1}) \}+A_{n-1} \}
\end{equation}
%=============================================================================
. Note that, during the n-th frame, the number of the transmitted packets have to be less than $Q_n$ and the number of packets can't exceed the maximum queue length $Q_{L}$.

We exploit an augmented FSMC to analyze the system \cite{LZ05QAM}. At first, we think about the FSMC model without the power control scheme. The state of the FSMC model is defined as a state pair $(Q_{n},C_{n})$. Let the $P_{\alpha, \beta}$ denote the probability of the state transition from state $\alpha$ to state $\beta$. For all $(q,c)\in (Q_{n},C_{n})$ and $(q',c')\in (Q_{n+1},C_{n+1})$, $P_{(q,c),(q',c')}$ is defined as
%=============================================================================
\begin{multline}
P_{(q,c),(q',c')} = \\
P(C_{n+1} = c' | C_{n} = c)P(Q_{n+1} = q' | Q_{n} = q, C_{n} = c)
\label{eqadfadif}
\end{multline}
%=============================================================================
. In (\ref{eqadfadif}), the $P(Q_{n+1} = q' | Q_{n} = q, C_{n} = c)$ is independent of power level of the femto AP. The $P(Q_{n+1} = q' | Q_{n} = q, C_{n} = c)$ can be calculated as (\ref{aaccdddcc}).
%========================================================
\begin{figure*}[!t]

\normalsize

%==========================================================
\begin{numcases}{P(Q_{n+1} = q' | Q_{n} = q, C_{n} = c)=} \nonumber
P(A_{n} = q' - max\{ 0, q-D(c) \},  & if $max\{ 0, q-D(c) \} \le q' < Q_{L}$ \\ \nonumber
0, & if $0 \le q' < max\{ 0, q-D(c) \}$ \\
1- \sum_{0 \le q' < Q_{L}} P(Q_{n+1} = q' | Q_{n} = q, C_{n} = c)  ,  & if $q'=Q_{L}$  \label{aaccdddcc}
\end{numcases}
%=========================================================

\hrulefill
\vspace*{4pt}

\end{figure*}
%========================================================
However, the $P(C_{n+1} = c' | C_{n} = c)$ in (\ref{eqadfadif}), which is the channel state transition probability, is  changed as the average SINR is changed, since the probability is dependent on the average SINR \cite{LZ05QAM}. Therefore, we have to make two state transition matrices for each mode because the femto AP has two level of transmission power. Let $\bm{P_{1}}$ and $\bm{P_{2}}$ denote the state transition matrices for mode1 and mode2,  respectively. Since the femto AP uses half power to transmit packets in the case of mode1, the $\bm{P_{1}}$ can be constructed by the Eq.(\ref{eqadfadif}) where the average SINR is $\bar{\gamma}-3 (dB)$. Simillary, The $\bm{P_{1}}$ is also calculated by the Eq.(\ref{eqadfadif}) where the average SINR is $\bar{\gamma} (dB)$.

%================================================================
\begin{figure}[!t]
\centering{\includegraphics[width=3.3in]{fig/frameblock}}
\caption{Frame and frame block} \label{frameblock}
\end{figure}
%================================================================

We need a stationary probability which contains the probability about modes, so that we could analyze system performances such as power consumption and everage packet drop rate. Therefore, we have to redefine the state pair to contain the mode state. According to our proposed scheme, the femto AP changes its transmission power in $N_{L}$ frames interval. So, we make frame block concept in order to define the state pair. As shown in Fig.\ref{frameblock}, $N_L$ frames make one frame block. Let $M_{N}$ denote the mode number at the N-th frame block. Then, the state pair can be constructed as $(M_{N}, Q_{N \times N_{L}}, C_{N \times N_{L}})$. By using the state pair for frame block, we can make an FSMC model.

Since the operation mode is not changed during a frame block, the queue length and the channel state are changed by the transition matrices $\bm{P_{1}}$ during the frame block when the operation mode is mode1. Simillary, if the operation mode is mode 2, the queue length and the channel state are changed by the transition matrices $\bm{P_{2}}$ during the frame block. Since $P^n$ means the state transition probability after n frames, we can calculate the probability of state transition after a frame block when we use $\bm{P_{1}}^{N_{L}}$ and $\bm{P_{2}}^{N_{L}}$. According to proposed scheme, when the femto AP determine operation mode, the femto AP can't have more than $\beta$ packets in the queue if the power mode becomes mode1 and the queue length have to be less than $\alpha$ when the power mode becomes mode2. Therefore, if $q \ge \beta$, $(1,q,c)\notin (M_{N}, Q_{N \times N_{L}}, C_{N \times N_{L}})$ and if $q < \alpha$, $(2,q,c)\notin (M_{N}, Q_{N \times N_{L}}, C_{N \times N_{L}})$. 

The transition probability from $(1,q,c)$ to $(1,q',c')$ is expressed as
%===========================================================================
\begin{equation}
\bm{P}_{(1,q,c),(1,q',c')} = {\bm{P_{1}}^{N_{L}}}_{(q,c),(q',c')}
\label{fgdfvbvsdf}
\end{equation}
%========================================================================
. When the mode changes from 1 to 2, the channel state also changes. Since the transit power increases, the channel state is added by one except the case that the channel state is already the maximum number 9. Therefore, the transition probability is described as follows.
%==========================================================
\begin{numcases}{\bm{P}_{(1,q,c),(2,q',c')}=} \nonumber
{\bm{P_{1}}^{N_{L}}}_{(q,c),(q',c'-1)},  & if $1<c'<9$ \\ 
0, & if $c'=1$ \label{vedzd}
\end{numcases}
%=========================================================
%=============================================================================
\begin{equation}
\bm{P}_{(1,q,c),(2,q',9)}= {\bm{P_{1}}^{N_{L}}}_{(q,c),(q',8)} + {\bm{P_{1}}^{N_{L}}}_{(q,c),(q',9)}
\label{bdfvdf}
\end{equation}
%=============================================================================
Simillary, the transition probability from $(2,q,c)$ to $(2,q',c')$ is expressed as
%==========================================================================
\begin{equation}
\bm{P}_{(2,q,c),(2,q',c')} = {\bm{P_{2}}^{N_{L}}}_{(q,c),(q',c')}
\label{adgbvvde}
\end{equation}
%==========================================================================
. When the mode is changed from 1 to 2, the femto AP increases its transmission power by 3dB. Thus, the transition probability from (2,q,c) to (1,q',c') is represented as follows by considering the channel state changing according to the mode transition.
%==========================================================
\begin{numcases}{\bm{P}_{(2,q,c),(1,q',c')}=} \nonumber
{{\bm{P_{2}}^{N_{L}}}_{(q,c),(q',c'+1)}},  & if $1<c'<9$ \\ 
0, & if $c'=9$
\label{aaeeebbb}
\end{numcases}
%=========================================================
%=============================================================================
\begin{equation}
\bm{P}_{(2,q,c),(1,q',1)}= {\bm{P_{1}}^{N_{L}}}_{(q,c),(q',1)} + {\bm{P_{1}}^{N_{L}}}_{(q,c),(q',2)}
\label{bdfvdfddfd}
\end{equation}
%=============================================================================
By using the Eq. (\ref{fgdfvbvsdf}) through (\ref{bdfvdfddfd}), the state transition matrix is defined as Eq.(\ref{tretwr}). Since the Markov chain which is defined by the transition matrix $\bm{P}$ is finite, homogeneous, and irreducible, the markov has an unique stationary distribution $\pi$ \cite{markov}. 
%========================================================
\begin{figure*}[!t]

\normalsize
%===========================================================================
\begin{equation}
\bm{P} = 
\begin{bmatrix}
P_{(1,0,1),(1,0,1)} & \cdots&P_{(1,0,1),(1,0,9)} & \cdots & P_{(1,0,1),(1,\beta -1,9)}& P_{(1,0,1),(2,\alpha,1)}& \cdots & P_{(1,0,1),(2,Q_{L},9)}\\
\vdots & \ddots & \vdots & \ddots & \vdots & \vdots &\ddots &\vdots \\
P_{(1,0,9),(1,0,1)}& \cdots& P_{(1,0,9),(1,0,9)}& \cdots& P_{(1,0,9),(1,\beta -1,9)}& P_{(1,0,9),(2,\alpha,1)}&\cdots&P_{(1,0,9),(2,Q_{L},9)}\\
\vdots & \ddots & \vdots & \ddots & \vdots & \vdots &\ddots &\vdots \\
P_{(1,\beta -1,9),(1,0,1)} & \cdots& P_{(1,\beta -1,9),(1,0,9)}& \cdots& P_{(1,\beta -1,9),(1,\beta -1,9)}& P_{(1,\beta -1,9),(2,\alpha,1)}&\cdots&P_{(1,\beta -1,9),(2,Q_{L},9)}\\
P_{(2,\alpha,1),(1,0,1)} & \cdots& P_{(2,\alpha,1),(1,0,9)}& \cdots& P_{(2,\alpha,1),(1,\beta -1,9)}& P_{(2,\alpha,1),(2,\alpha,1)}&\cdots&P_{(2,\alpha,1),(2,Q_{L},9)}\\
\vdots & \ddots & \vdots & \ddots & \vdots & \vdots &\ddots &\vdots \\
P_{(2,Q_{L},9),(1,0,1)} & \cdots& P_{(2,Q_{L},9),(1,0,9)}& \cdots& P_{(2,Q_{L},9),(1,\beta -1,9)}& P_{(2,Q_{L},9),(2,\alpha,1)}&\cdots&P_{(2,Q_{L},9),(2,Q_{L},9)}\\
\label{tretwr}
\end{bmatrix}
\end{equation}
%===========================================================================
\hrulefill
\vspace*{4pt}
\end{figure*}
%========================================================
Then, the stationary distribution $\pi$ satisfies the equality
%=============================================================================
\begin{equation}
\pi = \pi \bm{P}
\end{equation}
%=============================================================================
. If we know the stationary distribution, we could estimate the system performance. Fortunately, the stationary probability can be calculated by using computer programs.

In order to analyze the system performance by using the stationary distribution $\pi$, we shold split the $\pi$ by the operation mode because the state transition during a flame block has dependent on the mode. Let $\pi_{1}$ and $\pi_{2}$  denote the fraction of the $\pi$ for mode1 and mode2, respectively. Then, the $\pi_{1}$ is defined for all $(Q_{n} , C_{n})$ as follows 
%================================================================
\begin{equation}
\pi_{1} = [{\pi_{1}}_{(0,1)},\cdots ,{\pi_{1}}_{(0,9)},\cdots ,{\pi_{1}}_{(Q_{L},1)},\cdots ,{\pi_{1}}_{(Q_{L},9)}]
\end{equation}
%================================================================
, where the ${\pi_{1}}_{(q,c)}$ can be expressed as
%==========================================================
\begin{numcases}{ {\pi_{1}}_{(q,c)}= } \nonumber
   \pi_{(1,q,c)},  & for $q < \beta$ \\
    0,  & for otherwise  \label{pa}
\end{numcases}
%=========================================================
. Similary, the $\pi_{2}$ is also described as
%================================================================
\begin{equation}
\pi_{1} = [{\pi_{2}}_{(0,1)},\cdots ,{\pi_{2}}_{(0,9)},\cdots ,{\pi_{2}}_{(Q_{L},1)},\cdots ,{\pi_{2}}_{(Q_{L},9)}] \label{thfgw}
\end{equation}
%================================================================
%==========================================================
\begin{numcases}{ {\pi_{2}}_{(q,c)}= } \nonumber
    0,  & for $q < \alpha$ \\ 
    \pi_{(2,q,c)},  & for otherwise  \label{gefdvcvd}
\end{numcases}
%=========================================================
.

We can exploit the $\pi_{1}$ and $\pi_{2}$ to calculate the packet drop probability due to the overflow of the queue. The packet drops can be appeared at every frame although the stationary distribution is calculated for frame blocks. Therefore, when we calculate the expected number of dropped packets, we have to sum the dropped packets during a frame block. Let $D_{V(q,c)}$ denote the expected number of dropped packets at state (q,c) and $D_V$ denote the vector that is consist of the $D_{V(q,c)}$. Then, the $D_V$ is represented as
%================================================================
\begin{equation}
D_{V} = [D_{V(0,1)},\cdots ,D_{V(0,9)},\cdots ,D_{V(Q_{L},1)},\cdots ,D_{V(Q_{L},9)}]^{T} \label{thfgw}
\end{equation}
%================================================================
and the $D_{V(q,c)}$ is calculated as Eq.(\ref{dagaggad}).
%========================================================
\begin{figure*}[!t]

\normalsize

\begin{equation}
\label{dagaggad} 
D_{V(q,c)} = \sum_{a\ge 0}[max\{0,a-Q_{L}+max\{0,q-MCS(c)\} \}\times P(A=a)] 
\end{equation}

\hrulefill
\vspace*{4pt}

\end{figure*}
%========================================================
By using the vector $D_{V}$, we can obtain the expected number of the dropped packets during one frame block as 
%=========================================================
\begin{equation}
E\{Drop\}=\sum^{2}_{j=1} \pi_{j} \times ( \sum^{N_{L}-1}_{i=0}P^{i}_{j} ) \times D_{V}
\end{equation}
%=========================================================
, since, during the frame block, the stationary probabilities $\pi_{1}$ and $\pi_{2}$ are varied with transition matrix $P_{1}$ and $P_{2}$ respectively. Finally, we can derive the packet dropping probability and the expected throughput as follows, respectively.
%========================================================
\begin{equation}
\label{fadfa}
P_{d}= \frac{E\{Drop\}}{\lambda N_{L}} 
\end{equation}
%=========================================================
\begin{eqnarray}\nonumber
E\{Throughput\}&=&(\lambda )(1-P_{d})\\
               &=& \lambda - \frac{E\{Drop\}}{N_L}
\label{ogkvdie}
\end{eqnarray}
%========================================================


\hfill

\section{Numerical Results}\label{sec:numerical-results}
In previous section, we make a system model by using FSMC model. In this section we explain the numerical results. The parameters, which are used for the numerical results, are defined in Table. \ref{parameters}.

\begin{table}[!t]
\renewcommand{\arraystretch}{1.2}
\caption{Parameters for numerical results}
\label{parameters}
\begin{center}
\begin{tabular}{|c|c|}
\hline
Parameter& Value \\
\hline \hline
 Average SINR $\Bar{\gamma}$& 20dB\\ 
\hline
 Frame block length $N_{L}$& 5frames\\
\hline
 Maximum queue length $Q_L$& 100packets\\
\hline
 Nakagami parameter m& 1\\
\hline
 Frame duration $T_f$& 5ms\\
\hline
 Doppler frequency $f_d$& 4Hz\\
\hline
\end{tabular}
\end{center}
\end{table}
%================================================================
\begin{figure}[!t]
\centering{\includegraphics[width=3.3in]{fig/results3}}
\caption{Average throughput vs traffic density} \label{result1}
\end{figure}
%================================================================
%================================================================
\begin{figure}[!t]
\centering{\includegraphics[width=3.3in]{fig/results2}}
\caption{Transmission power vs traffic density} \label{result2}
\end{figure}
%================================================================

Fig. \ref{result1} represents the average throughput as traffic density. It is shown that our proposed scheme has almost the same data rate as the conventional scheme which uses constant transmission power. Therefore, we can expect that the throughput of the femto cell doesn't decrease even if the femto AP uses the proposed scheme.

We show the probability of the mode1 so that we could get results for power consumption of the femto AP. Since the femto AP uses half power to transmit a frame in mode1, the probability of the mode1 means power saving factor. Fig. \ref{result2} shows the probability of mode1 as traffic density. As you can see, the probability decreases as the traffic density increases. Therefore, we can make conclusions that the proposed scheme can reduce its transmission power without any throughput loss since the proposed scheme changes the transmission power according to traffic density and the proposed scheme can reduce interference to neighbor cells since it reduces its transmission power.


\hfill


\section{Conclusions}\label{sec:conclusions-1}
In this paper, we have suggested a solution for femto cell interference issues. The proposed scheme could reduce the downlink interference from a femto AP to neighbor cell users without throughput loss because it controls a femto AP's transmission power adaptively according to the traffic density of the femto AP. Moreover, based on the FSMC model \cite{LZ05QAM}, we analyze the performance of proposed system. Since our proposed scheme change the transmission power of the femto AP in $N_L$ frames interval, we introduce a fame block concept and construct FSMC model with the frame block. Therefore, we can analyze our proposed scheme though the scheme changes the transmission power according to the traffic load of the femto AP.

There are some remaining challenges for femto AP's power control. Since we just show the power reduction of femto AP, We should analyze throughput enhancement of neighbor nodes. Moreover, we should study more powerful scheme than our proposed one.


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